Earthquake-induced sloshing effects on the hydrodynamic pressure response of rigid cylindrical liquid storage tanks using CFD simulation

Engineering Structures 197 (2019) 109376

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Earthquake-induced sloshing effects on the hydrodynamic pressure response of rigid cylindrical liquid storage tanks using CFD simulation Tae-Won Kanga, Hyun-Ik Yangb, Jong-Su Jeonc,

T

⁎

a

Department of Mechanical Design Engineering, Hanyang University, Seongdong-gu, Seoul 04763, Republic of Korea Department of Mechanical Engineering, Hanyang University, Ansan, Gyeonggi-do 15588, Republic of Korea c Department of Civil and Environmental Engineering, Hanyang University, Seongdong-gu, Seoul 04763, Republic of Korea b

A R T I C LE I N FO

A B S T R A C T

Keywords: Earthquake-induced liquid sloshing Computational fluid dynamics Rigid liquid storage tank Vertical ground motion Water elevation Tank dimensions

This study aims at evaluating nonlinear hydrodynamic pressures on the inner wall of rigid cylindrical tanks, resulting from liquid sloshing due to the simultaneous action of two horizontal and one vertical ground motion components. For this purpose, the computational model that can predict realistic dynamic fluid responses is validated through comparison of simulated and experimental results reported in the literature. Using the validated computational fluid dynamics modeling technique, the computational model of a prototype cylindrical steel water tank constructed in California is developed to examine the effect of vertical ground motion component on the hydrodynamic pressure response of the tank. Comparison with and without vertical excitation indicates that the vertical motion significantly increases the hydrodynamic pressure on the inner wall of the tanks. Additionally, the effect of water elevation and tank dimensions (height and diameter) on the hydrodynamic pressure is investigated when the tanks are simultaneously subjected to two horizontal and one vertical ground motion components. As the water level and tank dimensions increase, the peak hydrodynamic pressure increases due to the increase of liquid mass.

1. Introduction Above ground liquid storage tanks have suffered serious damage during historical seismic excitations. Especially, the violent motion of liquid in partially filled tanks induced by earthquakes may result in severe sloshing loads on the structural system, leading to the local yielding, buckling, and unrepairable collapse of the tanks. As shown in past earthquakes (the 1964 Alaska, the 1964 Niigata, the 1979 Imperial Valley, the 1989 Loma Prieta, and the 1994 Northridge earthquake) [1–4], the tank failure and resulting liquid spillage may lead to the adverse of economic and environmental impacts on the earthquake affected area. Thus, it is very important to quantify the forces due to earthquake-induced sloshing in the design of liquid storage tanks. An early attempt to evaluate the hydrodynamic loads of liquid storage tanks due to earthquakes is found in the work of Housner [5]. The hydrodynamic pressure loading is assumed to consist of a convective component resulting from sloshing motion of liquid at the low frequency range and an impulsive component caused by the lateral motion of the tank and participating liquid at the high frequency range. However, the Housner method is based on numerous assumptions that cannot be applied to general cases, especially use of linear sloshing

⁎

assumption. This restriction may underestimate the structural response of tanks. Chen et al. [6] developed an implicit finite difference approach to simulate large amplitude sloshing motion of liquid under harmonic excitations and earthquakes. They concluded that nonlinear sloshing effects should be accounted for in the seismic design of tanks because the amplitude due to the nonlinear sloshing exceeds linear predictions. Although the seismic response of liquid storage tanks subjected to horizontal ground motions have been assessed by several researchers [7–10], studies on the seismic vulnerability assessment of the tanks under vertical ground motions coupled with horizontal ground motions are yet scarce. However, as stated by Haroun and Tayel [11], the vertical excitation in liquid storage tanks can be transmitted into horizontal dynamic loads, and thus the wall of the tanks may suffer radial deformations along with axial deformations. Haroun and Tayel [11] computed the tank response under the simultaneous action of both vertical and lateral earthquake excitations, and then evaluated the relative significance of the vertical component on the overall seismic behavior of liquid storage tanks. They found that dynamic hoop stresses due to vertical excitations are comparable to the hydrostatic hoop stresses. However, they neglected liquid sloshing in the simulations and

Corresponding author. E-mail addresses: [email protected] (T.-W. Kang), [email protected] (H.-I. Yang), [email protected] (J.-S. Jeon).

https://doi.org/10.1016/j.engstruct.2019.109376 Received 24 November 2018; Received in revised form 24 May 2019; Accepted 2 July 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

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nonlinear fluid, the structural response may be overestimated due to excessive fluid pressure.

reflected only the impulsive and the short-period pressure components. Kianoush and Chen [12] examined the dynamic response of rectangular concrete liquid storage tanks under horizontal and vertical ground acceleration to highlight the effect of vertical earthquake component on the global response of the tanks. It was concluded that the presence of the vertical component significantly affects the response of the tank wall. However, their analytical model did not explicitly account for the sloshing due to the vertical excitation. Ghaemmaghami and Kianoush [13] introduced the finite element method to examine the dynamic behavior of rectangular shallow and tall containers in a two dimension under both horizontal and vertical ground motions. They indicated that (1) the vertical excitations may lead to an increase in the convective response of the system, but does not significantly affect the impulsive behavior and (2) the effect of vertical acceleration on the dynamic response of the liquid tanks is less significant when horizontal and vertical ground motions are simultaneously applied. Yamada et al. [14] examined the influence of nonlinear boundary conditions at the liquid surface on the sloshing heights in cylindrical tanks under horizontal and vertical ground motion. Kim et al. [15] developed analytical solutions to evaluate dynamic response characteristics of partially filled flexible rectangular fluid containers subjected to horizontal or vertical ground excitations. However, the above two studies [14,15] did not directly account for the sloshing motion at the free surface resulting from the mixture between air and fluid. Since all of the above studies have focused mainly on the structural response of liquid storage tanks under vertical ground motions as well as accounted for partial nonlinear sloshing phenomenon, they cannot fully address the effect of nonlinear sloshing due to the flow of liquid in the tanks. To more realistically capture the nonlinear behavior of fluid sloshing in liquid storage tanks, numerous studies have currently employed computational fluid dynamics (CFD) analysis. Vakilaadsarabi et al. [16] used the CFD simulation to examine sloshing effects in liquid storage tanks due to long period ground motions using unidirectional lateral excitations. Their work revealed that the severity of sloshing and resulting dynamic pressure loads depends on the tank geometry, the depth of the liquid, and the amplitude and the nature of the tank motions. Godderidge et al. [17] investigated the effect of sloshing on the wall of rectangular containers under unidirectional lateral oscillations using the inhomogeneous model and homogeneous model in CFD simulations. Frandsen [18] developed a nonlinear finite difference model for some numerical examples on the basis of inviscid flow equations. Horizontal and vertical harmonic excitations are applied to a two-dimensional tank model to simulate sloshing wave motion. However, the imposed input motions are relatively simple (not realistic) in the sloshing problem induced by seismic excitations. In summary, the above studies related to the sloshing problem in CFD analyses did not employs realistic ground motion histories, especially the inclusion of vertical ground motions, and did not address the validation of the numerical model through the comparison of experimental results such as shaking table tests. In this study, CFD is used to analyze the nonlinear characteristics of the pressure acting on the inner wall of rigid cylindrical steel tanks due to three orthogonal components of earthquakes (two horizontal components and one vertical component). To achieve this goal, the computational model for the tank is validated in ANSYS-CFX [19] using existing experimental data. The validated modeling approach is implemented into the computational model of a real steel tank located in California. The effect of the vertical component of ground motions on the hydrodynamic pressure response acting on the tank wall is examined. Additionally, the influence of water elevation and tank dimensions on the hydrodynamic pressure response acting on the rigid tank wall is investigated. The CFD modeling techniques and numerical observations of this study provide a basis for further work to perform the fluid-structure interaction (FSI) analysis for flexible tanks. If the FSI analysis for the realistic (flexible) tank wall is performed without satisfying the numerical convergence to predict the flow change of the

2. Governing equations and model validation In general, CFD is based on the Reynolds-averaged Navier-Stokes model (RANS) model. RANS can be divided into Reynolds stress model and eddy viscous turbulence model by taking momentum averaged over time. The standard model commonly used in turbulence models is the k–ε model, since it has a capability of finding a robust and reasonable solution to a wide range of flow. However, when approaching to nonequilibrium boundary layers, the k–ε model reaches its limit. The problem of predicting flow near the wall has an effective solution, but the separation of flow is far from the actual behavior. Although another turbulence model, k–ω model, is ineffective due to its large mesh dependence near the wall, it reasonably solves the turbulent shear stress transfer problem and finds a good tendency to account for the flow separation problem. Thus, both the turbulence models may not realistically predict the dynamic fluid response in the non-equilibrium flow problem. To alleviate these issues, this study adopts the shear stress transport (SST) model [20]. The SST model has the advantages of the kε and k-ω models (the merits of the flow near the wall and the flow in the state separated from the wall). Additionally, this study employs the homogenous model with computational efficiency to simulate the interface problem of free surface model between the air and liquid. 2.1. Governing equation 2.1.1. Continuity and momentum equation N Assuming that the total volume of fraction is ∑α p γα = 1, the conservation equation is incompressibility, and ∑α

∂ (γ U ) ∂xj α αj

= 0 between

the phases, the continuity and momentum equations of multiphase flow can be described as Eqs. (1) and (2) [21].

∂ ∂ (γ ρ ) + (γ ρ Ui ) = Γα , Γα = ∂t α α ∂x i α α

Np

∑β=1 Γαβ

(1)

∂ ∂ (γ ρ Ui ) + (γ ρ Ui Uj ) ∂t α α ∂x j α α = −γα

∂Uj ⎞ ⎤ ∂p ∂ ⎡ ∂U γ μ ⎛⎜ i + + + ⎟ ∂x i ∂x j ⎢ α α ⎝ ∂x j ∂x i ⎠ ⎥ ⎣ ⎦

Np

∑ (Γαβ Uiβ − Γβα Uiα )+Fb β=1

(2) where ρ is the density in kg/m3, γ is the fixed volume of fraction, p is the pressure in Pa, and Ui is the velocity in m/s. μα

(

∂Ui ∂xj

+

∂Uj ∂xi

)=τ

ij

is

the stress tensor in Pa, where i , j contain tensors. Fb is the body force in N, which acts on the fluid inside the volume and along the gravitational acceleration g in m/s2, Γαβ is the mass transfer between the phases. Eqs. (1) and (2) represent the homogeneous model.

Γαβ = ṁ αβ Aαβ

(3)

where ṁ αβ of the mass transfer (in kg/s) is the mass flow rate per interfacial area from the phase α and β , Aαβ is proportional to the volume fraction density and interfacial area density between the phases, and Aαβ = |∇γα|. Note that the velocity field should be the same between the two phases, and the conservation of the momentum scheme should be solved only for one set. In the case of the homogeneous model, partial differential equations (PDEs) are used to calculate the interphase transfer terms, and thus the continuity and momentum equations (Eqs. (1) and (2)) are newly derived. The volume integration with the diffusion and gradient operators must be converted to the surface integration. The continuity (Eq. (4)) and momentum equations (Eq. (5)) are defined by applying the Gaussian Divergence Theorem as the integral over S = dΩ can be written as the integral over the volume Ω. Therefore, continuity and mass conservation can also be written in the 2

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accuracy and flow of the pressure gradient. The baseline model [20] combines the advantages of the willocox and the k–ε model, but cannot estimate the surface flow. Menter [20] details the SST model that combines the advantages of both models. The eddy-viscosity of the SST model can be written as:

vt =

a1 k max(a1 ω, SF2)

F2 = tanh(arg22),

PDE form:

∫ γα ρα dΩ + ∮ γα (ρα Ui)·ndS = ∫ (Γαβ Uiβ − Γβα Uiα ) dΩ

d dt

∫ γα ρα Ui dΩ + ∮ γα ρα Ui (Uj·n) dS = − ∮ γα PndS

Ω

Ω

(4)

Ω

s

+

∮ γα μα ⎛ ∂∂Uxji

+

Γβα Uiα ) dΩ

∫ ρα gdΩ

⎜

s

−

∂Uj ⎞

⎝

+

⎟

∂x i ⎠

ndS +

∫ (Γαβ Uiβ Ω

Ω

d dt

(5)

where the body force in N is defined as ∫ ρα gdΩ, which acts on the fluid

∫ ρφdΩ − ∮ ρφ (us − Vgrid)·ndS = ∮ Γdisp∇δ·ndS + ∫ SΦ dΩ Ω

s

s

inside the volume integral and generally includes the gravity and surface tension. The surface tension is the total force on the control volume due to the interfacial tension. This study uses a tension coefficient of 0.072 N/m following the work of Brackbill et al. [28].

whereφ is the conserved quantity per unit mass and Φ is the conserved quantity per unit volume. Thus, Φ = ρφ . Vgrid represents the velocity at which a mesh element is moved by the external excitation. Eq. (10) guarantees the safety of the numerical solution through the implicit second order backward Euler equation of Eq. (7). The first term in the right hand side in Eq. (10) is the diffusion term that controls the movement by the external excitation. This term is called the displacement diffusion equation (DDE), which represents the moving boundary of the mesh induced by the excitation [19]. DDE can be written in Eq. (11). SΦ includes the pressure and body forces. The sources are assumed to be specified per unit volume of geometry, not per unit volume.

2.1.2. Element implementation The area where the flow is calculated is represented by the node point in Fig. 1. The discretization of the governing equations requires the integration point of the control volume associated with each node [22]. The element uses the vertex centered (node based) method instead of the cell centered method to construct a sub-element divided by the dotted lines and transmits the data to each node. A high resolution scheme based on the boundary principle presented by Barth and Jesperson [23] is introduced to allow the second order accuracy of discretization for the advection term, including that under-shoot or over-shoot can be reduced for each integration point around the node. The high resolution scheme can be expressed as:

Cstiff

∀ref ⎞ ∇ ·(Γdisp ∇δ ) = 0, Γdisp = ⎛ ⎝ ∀ ⎠ ⎜

∫ γα ρα ΦdV ≈ ΔVt ⎡⎣ 32 (γα ρα Φ)n − 2(γα ρα Φ)n−1 + 12 (γα ρα Φ)n−2⎤⎦ v

⎟

(11)

where δ is the displacement relative to the previous mesh locations and Γdisp is called the mesh stiffness.∀ref is the reference volume, ∀ is the control volume size, and Cstiff is the stiffness model exponent. To control the mesh stiffness, the stiffness values between 10−15 and 1015 must be defined. The relationship between the motion and stiffness must be tested; the sudden change of the stiffness due to the sudden movement in a short time period cannot ensure the numerical convergence. In the case of an internal fluid-domain with a moving boundary method via DDE, it is possible to further guarantee the convergence for the fluctuation of the nonlinear internal flow during large deformations because the mesh can be controlled to be distorted or deformed as the mesh changes. Thus, this study selects the displacement time history as input excitations instead of the acceleration time history. This moving mesh technique is useful because it guarantees the mesh quality when considering the boundary couples of the flexible wall and the fluid domain and ensures sufficient numerical stabilities. However, the boundary conditions for the fluid-structure interaction are not considered in this study because the wall is assumed to be rigid.

(6)

where the node value for β is regarded as the minimum value of all the integration points surrounding the nodes. Note that the value β should not exceed one. Eq. (6) is used to reduce the number of oscillations, and the accuracy of convergence is improved by using the second order backward Euler equation [19]:

d dt

Ω

(10)

Ω

→ Φip = Φup + β∇Φ·R

(9)

2.1.4. Transport equation considering deformation control volumes method General conservation equations with deformation control volumes in the transport equation can be expressed as the transient, convection, diffusion, and source term:

s

s

k 500ν arg2 = max ⎜⎛2 , 2 ⎞⎟ ⎝ 0.09ωy y ω ⎠

In the equation, F2 is defined as a blending function based on the distance to the surface and the flow variable, which is a mixing function similar to F1 [19]. F1 is based on the distance to the nearest surface and on the flow variables, which limits the constraint to the wall boundary layer, because the basic assumption about the free shear flow is incorrect. vt is the eddy viscosity in m2/s, k is the turbulent kinetic energy in m3/s2, a1 is a constant, ω is the turbulent frequency in Hz, S is the absolute value of the vorticity ∂u/ ∂y , y is the distance in m to the next surface

Fig. 1. Integration points and nodes.

d dt

(8)

(7)

It is necessary to reduce the numerical error during iterations between the air and water phases. To guarantee convergence over time intervals, it is required to solve the problem of nonlinear flow by performing a repetitive calculation process at least 10 times during analysis.

2.2. Validation of computational model

2.1.3. SST turbulence model The SST model based k–ε model describes the turbulent shear stress transfer and is a model that can more accurately predict the start-up

In the design of liquid storage tanks, the movement of water surface 3

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1,000

1,000

Pressure meters

P4~P6

800

Loading direction P7~P9 Pressure meters

20 mm thick

Plan

P3

P9, P12

P6

P2

P8, P11

P5

P1

P7, P10

P4

400

[Unit: mm]

P1~P3

300

1,500

P10~P12

Loading direction

Elevation Fig. 2. Description of subject water tank used for model validation.

is assumed to be linear in the dynamic computational model. Thus, the model cannot account for the nonlinearity of the fluid motion due to external excitations. To more realistically obtain the internal pressure acting on the tank wall associated with earthquakes, the computational modeling approach of CFD is validated to compare simulated results with experimental data available in literature.

Additionally, the pressure gauges were installed according to the height of the four wall sides of the tank to analyze the hydrostatic pressure and mode shape along the height. The dynamic pressures measured from precision pore pressure sensors are used for this study to validate the computational modeling technique of CFD, and the model validation is described in the following section.

2.2.1. Description of steel water tank used for model validation To validate the computation model of a liquid storage tank, a steel water tank tested by Baek et al. [24] is selected for this study. Fig. 2 shows details of the tested specimen. The tested specimen is cylindrical; the diameter and height of the tank are 1 m and 1.5 m, respectively; and the height of water is 0.8 m. Because few shaking table test data exist that characterize the dynamic fluid response of tanks subjected to the horizontal and vertical excitations at the same time, this study selects the above specimen tested under a horizontal ground motion component. The input motion selected is the 1940 El Centro earthquake, NS component. The motion imposed in the shaking table is a 40% scale motion of the earthquake and its time step is 0.0039 s. Fig. 3 shows the acceleration and displacement time histories for the applied excitation. Its peak acceleration and displacement is 0.173 g and 11 mm, respectively. To more clearly examine the sloshing response of fluid in the tank, these authors fabricated the steel tank with a thickness of 20 mm, which ensures the rigid behavior of the tank. The nonlinear dynamic behavior of the fluid and its effect on the tank were evaluated using 12 pore pressure gauges, as shown in Fig. 2. Since the tested tank is 1.5 m high and the water level is 0.8 m, the water pressure acting on the inside is not so large, and thus a precision pore pressure meter capable of measuring micro-water pressure change was used. Three hydraulic pressure gauges were installed at 0 mm, 300 mm, and 700 mm height positions (distant from the bottom of the tank) on the inner wall surface in two horizontal directions (Fig. 2).

2.2.2. Model validation and governing equation of numerical implementation process The element model used in the CFD analysis is a hexahedral element based on a vertex centered (node based), and the elements of the cylindrical tank are represented by an O-grid type, as shown in Fig. 4, to maintain the alignment of grid shape. To keep the aspect ratio of the elements constant during the formation of the O-grid type element, the size of the element shape is adjusted similar to the ratio of the size of the tank. Fig. 5 shows the flowchart of calculating the sloshing phenomenon of fluid in the tank. When the seismic motion is applied to the bottom of the rigid tank shown in Fig. 2, the modeling parameters related to the fluid quantities are firstly initialized. The mesh stiffness between the nodes (Γdisp ) is then calculated through the mesh displacement diffusion equation with the regions of motion specification (Eq. (11)). If the stiffness between the nodes is too large, Eq. (11) does not converge. If the seismic motion generates the displacement of the mesh domain (∇δ ), the continuity and momentum equations (Eqs. (4) and (5)) are calculated by the high resolution scheme of Eq. (6) and the implicit second-backward Euler equation of Eq. (7). In this case, the water phase α and the air phase β are applied in the equations, and the turbulence is calculated using the SST model of Eqs. (8) and (9). The currently used VOF method depends on the interface capture method and is calculated when the interface is tracked across a fixed grid. The compressive discretization scheme maintains the interface boundary by solving the continuity transport equation for the volume fraction field.

(a)

(b)

Fig. 3. Input motions measured at the shaking table: (a) acceleration and (b) displacement time histories. 4

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nonlinear flow, at least 5 to 10 iterations should be performed. Thus, for every computational time, the iterative calculation process is required to more realistically capture the phenomenon of water movement. In addition, the lattice is constructed to form a boundary layer near the water level to properly predict the time step and the flow separation phenomenon. If the size of the element is too large, the pressure data acting on the wall would interpolate at the neighboring nodes, resulting in a slight difference from the pressure at the measuring point. Using the numerical implementation process described above, the model validation was performed. Table 1 presents the boundary conditions used in the CFD analysis. For the CFD analysis, this study imposes the displacement time history in Fig. 3(b) as the applied input motion, as used in Godderidge et al. [21]. Fig. 6 shows the comparison of hydrodynamic pressure time histories obtained from the CFD analysis and the shaking table test at the same positions. The figure includes the pressure response data of P1, P2, P4, and P5 in the loading direction. The response data near the free surface (P3 and P6) are excluded from the measurement due to the sensor error. In this study, the measured data are used, which exhibit the most hydrodynamic pressure along the loading direction. The initial responses at the beginning of the excitation are slightly different from the measured pressure responses because one second of zero ground motion acceleration is added in the beginning of the excitation to ensure the stability of numerical convergence. The model predicts the peak hydrodynamic pressure within a simulation error of 6%. Therefore, the numerical model in the CFD analysis is appropriate to capture the overall pressure response on the wall of the steel tank.

Fig. 4. O-grid type mesh of the cylindrical tank used in CFD analysis.

This is because a flow quantity in Φup in Eq. (6) can be obtained by interpolating the quantities computed at all nodes in an element (see Fig. 1). When the flow transports from Φup to Φip (in the direction of the → vector R ), the inter-node interpolation on the interface surface computes the flow at the integrating point (Eq. (6)). To predict the

Fig. 5. Sloshing analysis calculation process. 5

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Table 1 Setting conditions of CFD model. Parameter

CFD setting condition

Parameter

CFD setting condition

Water Air Turbulence model

Incompressible fluid Ideal gas SST model

Spatial discretization Scheme temporal discretization

Gradient dependent High Resolution Scheme Second order backward Euler scheme

(a) P1

(b) P2

(c) P4

(d) P5

Fig. 6. Comparison of experimental and simulated hydrodynamic pressure time histories.

19.507 m, respectively, and the tank is filled with water to a height of 17.557 m (water elevation level: 90% of the tank height). The tank rests on a 3.658 m deep concrete wall, and thus the bottom of the tank is assumed to be fixed. Fig. 7 also illustrates the location of pressure monitoring points and the loading direction of applied input excitations to be analyzed. Four points on each side of the wall (east, west, south, and north) are selected to monitor the earthquake-induced pressure acting on the wall of the tank. The monitoring points are equally distributed with an increment of one-fifth of the tank height from the bottom of the tank.

3. Description of real water tank and input ground motions 3.1. Description of real-scale water tank To examine the seismic pressure response of real tanks, a cylindrical steel tank employed in Haroun [25] is selected as the baseline structure for this study. Haroun [25] performed the ambient and forced vibration tests for this tank to better understand the dynamic characteristics such as natural frequency and mode shape. This tank is a welded steel, water storage tank owned by the Metropolitan Water District of Southern California. The tank is composed of a cylindrical thin shell with a height to diameter ratio slightly greater than one. As shown in Fig. 7, the thickness of the tank wall varies from 19 mm (at the bottom) to 6 mm (at the top). However, this study assumes the tank wall as rigid for CFD analysis. The diameter and height of the tank are 18.288 m and

3.2. Input ground motions To examine the seismic wall pressure of the real steel tank, three different earthquakes are extracted from the NGA-West2 on-line ground

Fig. 7. Configuration of the selected water tank, location of pressure monitoring points, and direction of input motions. 6

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(a) 1940 Imperial Valley earthquake

(b) 1995 Kobe earthquake

(c) 1971 San Fernando earthquake Fig. 8. Acceleration time histories of three earthquakes.

(a) 1940 ImperialValley earthquake

(b) 1995 Kobe earthquake

(c) 1971 San Fernando earthquake Fig. 9. Displacement time histories of three earthquakes.

7

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Table 2 Peak ground acceleration and displacement of the three earthquakes. Peak ground acceleration (g)

Peak ground displacement (mm)

Earthquake

Station

Longitudinal (L)

Transverse (T)

Vertical (V)

Longitudinal (L)

Transverse (T)

Vertical (V)

Imperial Valley Kobe San Fernando

El Centro Array #9 KJMA Pacoima Dam (upper left abut)

0.281 0.834 1.219

0.211 0.630 1.238

0.178 0.339 0.687

86.6 210.7 390.0

242.0 183.0 128.0

26.7 144.0 293.0

motions is more concentrated on the north side of the wall due to the coupled horizontal and vertical earthquake components. At each height level, the peak total pressure (maximum of four maximum total pressures at each height level) is governed by the pressure acting on the north side of the wall. The presence of vertical motion results in a 32%, 38%, 47%, and 64% increase in the peak total pressure at the monitoring points from α = 0.0 to α = 0.6, respectively. In the case of the Kobe earthquake, the inclusion of vertical earthquake component significantly increases the maximum total pressure on all four sides of the wall near the surface of water (54–76% increase), but has little or no influence on the pressure near the bottom of the tank. Thus, the difference in the maximum total pressure between adjacent monitoring points decreases as the normalized tank height increases in comparison to only use of two horizontal motions. The vertical component of this earthquake leads to changing the location on the circumference where the peak total pressure occurs; the peak total pressure under L + T motions is governed by the pressure acting on the east side of the wall while it under L + T + V motions is dominated by the pressure on the south side of the wall. This is due to the Coriolis effects resulting from the combined ground motion components. When the tank is subjected to two horizontal ground motion components, the internal fluid rotates in the horizontal plane with respect to the center. With the rotation, the centrifugal and Coriolis forces are generated as shown in Fig. 11. In the inertial frame of reference, the water particles move in a straight line along the transverse direction. However, in the view of the non-inertial frame of reference, the particles move following a curved path (opposite to the rotation direction) due to the Coriolis and centrifugal forces present in the frame. Also, the Coriolis effect on the wall due to the movement of the tank can be found in Eswaran and Saha [29]. Also, the presence of vertical motion results in a 1% and 66% increase in the peak total pressure at α = 0.0 and α = 0.6, respectively. Thus, the effect of vertical ground motion more significantly affects the pressure near the free surface than at the bottom of the tank. In the case of L + T + V motions of the San Fernando earthquake, the total pressure distribution along the tank height on the three wall sides (south, north, and east) is different from other two earthquakes; other two earthquakes reduce the maximum total pressure as the normalized tank height increases, while this earthquake increases the maximum total pressure as the normalized tank height increases from α = 0.2 to α = 0.6. The vertical ground motion significantly increases the maximum pressure near the surface of water on the above three walls (74–163% increase), but does not affect the maximum pressure near the free surface (α = 0.0) on the west side of the wall. However, since the peak pressure is mainly dominated by the pressure acting on the west side of the wall along all the height levels regardless of the presence of vertical earthquake component, the vertical ground motion has a minimal effect on the peak pressure (3–13% increase). In summary, the pressure response on the tank wall is significantly dependent on the ground motion characteristics and the presence of vertical ground motion component. In addition, the inclusion of vertical ground motion can change the response characteristics such as (1) the maximum pressure, (2) the maximum total pressure distribution along the tank height, and (3) the location on the circumference where the peak total pressure occurs. For these reasons, the hydrodynamic pressure (convective and impulsive component) resulting from the vertical ground

motion database [26]: the 1940 Imperial Valley, the 1995 Kobe, and the 1971 San Fernando earthquake. All earthquakes include three orthogonal components: two horizontal components and one vertical component. Especially, the ground motion data measured at the Pacoima Dam station recorded during the 1971 San Fernando earthquake are selected because this station is closest to the location of the selected steel tank. Also, this earthquake is one of the strongest earthquakes in the world and its vertical ground motion component is relatively large with respect to its horizontal components. Other two earthquakes are selected for the response comparison. The Kobe earthquake is a strong seismic event, but the vertical motion is relatively small compared to the horizontal components. The Imperial Valley earthquake is weaker than other earthquakes and its vertical component is very small. Figs. 8 and 9 show the acceleration and displacement time histories for each component of the three earthquakes, respectively. Table 2 summarizes the peak ground acceleration (g) and peak ground displacement (mm) for each component of the three earthquakes. As indicated in the model validation, displacement time histories of earthquakes are imposed to the bottom of the tank as the input excitations. 4. Significance of vertical ground motion As mentioned in Introduction, Haroun and Tayel [11] highlighted the importance of effects of vertical earthquake component on the seismic pressure response acting on the tank wall. To examine the pressure change inside the water tank due to the inclusion of vertical earthquake component, two types of input motions presented in Section 3.2 are selected in this study to perform CFD analyses: two horizontal earthquake components (L + T motions) versus two horizontal and one vertical earthquake components (L + T + V motions). Fig. 10 shows the maximum total pressure acting on the four (east, west, south, and north) sides of the tank wall along the normalized tank height (α ) under L + T motions and L + T + V motions of three earthquakes. Note that the total pressure is the summation of the hydrostatic and hydrodynamic pressures, and the normalized tank height is defined as the ratio of the height from the tank base to a pressure monitoring point to the tank height. The results without vertical ground motion shown in Fig. 10(a) indicate that the maximum total pressure is almost linearly distributed with respect to the normalized tank height. As shown in Fig. 10(b), as the normalized tank height (α ) increases, the maximum total pressure near the surface of water under L + T + V excitations increases for all wall sides in comparison to the case of L + T motions. Also, the inclusion of vertical motion reduces the difference in the maximum total pressure between adjacent monitoring points along the tank height. These observations are due to the fact that the vertical earthquake component affects the impulsive component of the fluid. In the case of the Imperial Valley earthquake, the inclusion of vertical earthquake component results in a 10% reduction in the maximum total pressure acting on the south side of the wall at the bottom of the tank, but significantly increases the maximum total pressure acting on the north side of the wall over the entire range of height (32–64% increase). In other words, the vertical ground motion increases a significant difference between the maximum total pressure on the north and other sides of the wall. This is associated with the fact that the pressure energy due to the combined lateral and vertical sloshing 8

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Imperial Valley EQ

Kobe EQ

San Fernando EQ

(a) Under L+T motions

Imperial Valley EQ

San Fernando EQ

Kobe EQ

(b) Under L+T+V motions Fig. 10. Comparison of maximum total pressure distribution under (a) L + T motions and (b) L + T + V motions of three earthquakes.

motion should be appropriately accounted for in designing the water tank. Figs. 12–14 plot the power spectral density (PSD) function (in g2/ Hz) of the input acceleration time histories shown in Fig. 7 and the PSD (in kPa2/Hz) of simulated hydrodynamic pressure responses on a specific side of the tank wall, which shows the peak hydrodynamic pressure acting on the normalized tank height (α ), under L + T and L + T + V components of three earthquakes. Typically, there exists a wide frequency gap between motions associated with the convective and impulsive effects. The convective effect results from the sloshing motion of water by the excitations, and thus is observed at the low frequency range of the PSD of the pressure response, which is similar to the frequency range of the PSD of the excitation. In contrast, the impulsive component due to the excitation usually lies in the high frequency range of the PSD of pressure response. Thus, it can be seen from the figures that the impulsive component significantly affects the energy of hydrodynamic pressure due to the Imperial Valley earthquake and the San Fernando earthquake (Fig. 12(a) and 14(a)), regardless of the presence of vertical earthquake component, while the convective component for the Kobe earthquake (Fig. 13(a)) more significantly

Fig. 11. Coriolis effect of internal fluid due to two horizontal earthquake ground motions.

(a)

(b)

(c)

Fig. 12. PSD due to the Imperial Valley earthquake: (a) PSD of input accelerations, (b) PSD of hydrodynamic pressure responses on the north wall under L + T motions, and (c) PSD of hydrodynamic pressure responses on the west wall under L + T + V motions. 9

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(a)

(b)

(c)

Fig. 13. PSD due to the Kobe earthquake: (a) PSD of input accelerations, (b) PSD of hydrodynamic pressure responses on the west wall under L + T motions, and (c) PSD of hydrodynamic pressure responses on the west wall under L + T + V motions.

(a)

(b)

(c)

Fig. 14. PSD due to the San Fernando earthquake: (a) PSD of input accelerations, (b) PSD of hydrodynamic pressure responses on the west wall under L + T motions, and (c) PSD of hydrodynamic pressure responses on the west wall under L + T + V motions.

reveals that there is a significant difference in the pressure response between the case of L + V motions and L + T + V motions of the Imperial Valley earthquake, although the vertical component is the smallest among three earthquakes. This finding is consistent with the PSD of the San Fernando earthquake with the largest vertical ground motion (Fig. 14(b) and (c)). Unlike the results of the above earthquakes, the Kobe earthquake (Fig. 13(b) and (c)) results in small hydrodynamic pressure energies in the high frequency range (impulsive component). The energy due to the vertical ground motion component is relatively small compared to other earthquakes. Thus, the pressure change along the normalized tank height (α ) is linear as shown in Fig. 10(a) and (b) because the Kobe earthquake depends on the convective component. The trend of PSD due to the input excitations of L + T + V motions of the Imperial Valley earthquake (Fig. 12(a)) is very similar to the trend of the energy of hydrodynamic pressure responses in the convective frequency range (about 0–6 Hz) (Fig. 12(c)). Also, in the region near 8 Hz, the energy of the vertical acceleration is larger than the energy of the two horizontal components. The PSD results for the hydrodynamic pressure response at this frequency range indicate that the pressure energy measured at the nearest to the water surface (α = 0.6) is higher than that at other height levels because the vertical ground motion component increases the impulsive component (high frequency range) (comparison of Fig. 12(b) and (c)) as well as the water surface elevation. The trend of PSD due to L + T motions of the San Fernando earthquake (Fig. 14(a)) is well captured in the PSD results of hydrodynamic pressure within the excited frequency range (Fig. 14(b)). It is observed that the hydrodynamic energy at the excited frequency range under L + T motions is higher than that under L + T + V motions; in this range, there is no vertical ground motion effect. However, the PSD results of the hydrodynamic pressure resulting from L + T + V motions match well with the trend of the PSD of the input excitations. At the nearest monitoring point to the water surface (α = 0.6), the hydrodynamic pressure energy under L + T + V motions is much higher than

impacts the energy of hydrodynamic pressure. This observation can be interpreted using the ground acceleration time history in the time domain. The energy of hydrodynamic pressure due to earthquakes depends on the shape of input ground motions, especially the duration of strong excitation. Acceleration time histories of the Imperial Valley earthquake (Fig. 8(a)) slowly decrease during a relatively long time (about 30 s) after reaching the peak ground acceleration. Acceleration time histories of the San Fernando earthquake (Fig. 8(c)) have large amplitudes during about 5 s before the peak ground acceleration. Thus, these earthquakes continuously provide inertia forces, which are applied to the movement of fluid in the tank, and thus can produce a large amount of hydrodynamic (impulsive component) pressure at the high frequency range. The impulsive component dominates the energy of hydrodynamic pressure. However, acceleration time histories of the Kobe earthquake (Fig. 8(b)) have small amplitudes before the peak ground acceleration, and suddenly decrease during a short time (about 10 s) after the peak ground acceleration. The significant decay of the input motions (a short duration of strong excitation) yields a small amount of the energy of impulsive component at the high frequency range, and thus the convective component governs the energy of hydrodynamic pressure. As shown in Fig. 12(a), the Imperial Valley earthquake has smaller ground accelerations and magnitude in comparison to the Kobe and San Fernando earthquakes. The energy for the input acceleration of two horizontal components is large up to about 8 Hz, but the energy of the vertical acceleration increases after 8 Hz. As shown in Fig. 13(a), the Kobe earthquake has large input acceleration energies within 4 Hz. Its vertical earthquake component has larger energy than the two horizontal components about 4–5 Hz. As plotted in Fig. 14(a), the San Fernando earthquake has the largest vertical component, and the energy of the vertical component is relatively larger than that of other two earthquakes. The energy of two horizontal components is widely distributed before 6 Hz, but the energy of the vertical component tends to increase in the vicinity of 8 Hz. The comparison of Fig. 12(b) and (c) 10

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that under L + T motions. This is due to a significant change in flow for the vertical movement of water induced by the vertical earthquake component. The gravitational inertia force resulting from the vertical ground motion component is added to the inertia force of the water particles under only horizontal components, resulting in the increase of the energy of hydrodynamic pressure near the water surface

Fernando earthquake (Fig. 17), the peak hydrodynamic pressure typically occurs on the east side of the wall irrespective of water elevation. The maximum pressure with the tank with FR = 50% is significantly reduced as the normalized tank height increases. However, the relation between the maximum pressure and normalized height is not linear for the tank with FR = 70% and FR = 90%. Especially, the maximum pressure on the east side of the wall at α = 0.0 (bottom of the tank) is slightly higher than that at α = 0.6 (close to the free surface). Unlike the results of Chen et al. [27] (FR = 50–70%), the results of this study indicate that overall the tank with FR = 90% exhibits the highest peak dynamic pressure among three cases of water elevation. This is due to the difference in the input excitation and tank shape between the two. To estimate pressure time histories on inner walls, Chen et al. [27] applied horizontal (unidirectional) harmonic excitations with the natural frequency of the fluid motion to rectangular tanks, while this study imposes recorded earthquake excitations with three orthogonal components to circular tanks.

5. Effect of water elevation level on hydrodynamic pressure response This section presents the effect of water elevation on the seismic pressure response of water tanks under L + T + V motions of three earthquakes. Here, the water elevation is defined as the ratio of the distance from the water surface from the bottom of the tank (Hw) to the tank height (in percentile). To achieve this goal, this study employs the baseline tank model shown in Fig. 7 with modification to the water elevation level. Three levels of water elevation (FR) are used: FR = 90% (Hw = 17.556 m), FR = 70% (Hw = 13.655 m), and FR = 50% (Hw = 9.754 m). The latter two elevation levels are determined following the work of Chen et al. [27], which stated that the sloshing effects on the hydrodynamic pressure response are significant for the elevation level ranging from 0.5 to 0.7. Figs. 15–17 show the maximum hydrodynamic pressure acting on the four sides of the tank wall with different levels of water elevation under L + T + V components of three earthquakes. In the figures, the pressure at α = 0.6 for the tank with FR = 50% exceeds the free water surface; non-zero value results from additional hydrodynamic pressure due to flow separation. It can be observed from the figures that as the water elevation level increases, the pressure near the surface of water increases. This is associated with the fact that as the water level decreases, an amount of water into which the air is entrained increases due to the excitations, and thus the fluid mass affecting the pressure on the wall decreases. In contrast, as the water level increases, the fluid mass affecting the pressure on the wall increases because the particles bounce off the tank wall, and thus the hydrodynamic pressure increases. In the case of the Imperial Valley earthquake (Fig. 15), the position on the circumference where the peak hydrodynamic pressure occurs is different for the tank with FR = 50% and FR = 70% (or FR = 90%); the peak hydrodynamic pressure for the tank with FR = 50% occurs on the east side of the wall while the peak hydrodynamic pressure for the tank with FR = 70% or 90% occurs on the north side of the wall. As mentioned before, this observation results from the Coriolis effects induced by simultaneously imposing three orthogonal components of the earthquake. Unlike the Imperial Valley earthquake, the Kobe earthquake does not change the position on the circumference where the peak hydrodynamic pressure occurs (Fig. 16). The trend of the maximum hydrodynamic pressure distribution over the tank height is similar regardless of the extent of water elevation. As the normalized height increases, the maximum hydrodynamic pressure acting on the east side of the wall decreases, but the pressure on other three sides of the wall slightly increases. For the San

(a) FR = 50%

6. Effect of tank dimensions on dynamic pressure response This section describes the influence of tank dimensions such as tank height and tank diameter on the hydrodynamic pressure of tanks subjected to L + T + V components of three earthquakes. As observed in the previous section, since the tank with FR = 90% generally exhibited the highest peak dynamic pressure among other cases of water elevation, this section only deals with numerical results of the tank with FR = 90%. 6.1. Tank height The baseline tank model shown in Fig. 7 is modified with two tank heights. Thus, three types of tank height are selected to examine the effect of tank height on the hydrodynamic pressure response of the tank wall: 0.75, 1, and 1.25 times the baseline tank height (H = 0.75Hbase = 14.630 m, H = 1.00Hbase = 19.507 m, and H = 1.25Hbase = 24.384 m). CFD analyses for the modified two models under L + T + V motions of three earthquakes are additionally performed in ANSYS-CFX [19]. Table 3 summarizes the peak hydrodynamic pressure at each normalized tank height for three tanks under three earthquakes. It is obvious that as the tank height increases, the peak hydrodynamic pressure increases. It can be explained that the increase of liquid mass results in the increase of internal impulsive component. When compared to the case of the baseline tank, the peak hydrodynamic pressure of the shorter tank decreases by about 9–29% and the peak hydrodynamic pressure of the taller tank increases by about 1–20%. When three tanks are subjected to the Kobe earthquake, the effect of tank height on the peak hydrodynamic pressure is the most significant. Figs. 18–20 show the PSD of hydrodynamic pressure response on a specific side of the wall for three tanks under three earthquakes. As the tank height increases, the energy of hydrodynamic pressure increases over the entire

(b) FR = 70%

(c) FR = 90%

Fig. 15. Maximum hydrodynamic pressure distribution with different levels of water elevation under the Imperial Valley earthquake. 11

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(a) FR = 50%

(b) FR = 70%

(c) FR = 90%

Fig. 16. Maximum hydrodynamic pressure distribution with different levels of water elevation under the Kobe earthquake.

range of frequencies. Especially, the Imperial Valley earthquake and the San Fernando earthquake more significantly increase the pressure energy at the high frequency range (impulsive component), while the Kobe earthquake more considerably increases the pressure energy at the low frequency range (convective component).

Table 3 Peak hydrodynamic wall pressure (kPa) under L + T + V components with different levels of tank height. Earthquake

6.2. Tank diameter The baseline tank model shown in Fig. 7 is also modified with two tank diameters. Thus, three types of tank diameter are chosen to investigate the influence of tank diameter on the hydrodynamic pressure response of the tank wall: 0.8, 1, and 1.2 times the baseline tank diameter (D = 0.8Dbase = 14.630 m, D = 1.0Dbase = 18.288 m, and D = 1.2Dbase = 21.946 m). Table 4 presents the peak hydrodynamic pressure at each normalized tank height for three tanks under three earthquakes. As the tank diameter increases, the peak hydrodynamic pressure typically increases. When compared to the case of the baseline tank, the peak hydrodynamic pressure of the narrower tank decreases by about 13–31%. However, the difference in the peak hydrodynamic pressure between the wider tank and baseline tank is marginal (within 8% and 14%) when the tanks are subjected to the Imperial Valley earthquake and the Kobe earthquake, respectively. However, the peak hydrodynamic pressure of the wider tank under the San Fernando earthquake increases by approximately 15% and 27% at the closest position to the free surface and the bottom of the tank, respectively. Figs. 21–23 plot the PSD of hydrodynamic pressure response on a specific side of the wall for three tanks under three earthquakes. As the tank diameter increases, the energy of dynamic pressure increases over the entire range of frequencies because of the increase of liquid mass.

Normalized tank height (α ) 0.0

0.2

0.4

0.6

Imperial Valley

H = 0.75Hbase H = 1.00Hbase H = 1.25Hbase

262.7 309.7 309.0

256.7 299.2 302.8

238.8 268.0 288.2

199.5 221.5 257.0

Kobe

H = 0.75Hbase H = 1.00Hbase H = 1.25Hbase

183.1 228.0 272.9

152.0 209.5 228.4

133.0 187.7 208.8

113.5 156.9 183.4

San Fernando

H = 0.75Hbase H = 1.00Hbase H = 1.25Hbase

264.7 333.0 361.9

259.5 285.2 317.7

254.2 313.0 339.0

275.6 338.9 348.9

purpose, ANSYS-CFX program that can realistically capture the nonlinear dynamic fluid response is used in this study; the shear stress transport model that can predict both the flow near the wall and the separation of flow is used in the modeling of CFD. The computational modeling technique is validated by comparing the simulated results with experimental data available in the literature. The simulated hydrodynamic pressure responses are well correlated with observed shaking table test data. Using the proposed modeling technique, the computational model of a real cylindrical steel water tank located in California (regarded as the baseline model) is developed to examine the effect of vertical ground motion component (with and without vertical component) on the hydrodynamic pressure response on the inner wall of the tank. The structure is assumed to be rigid for CFD analysis. For dynamic analyses, three recorded earthquakes are selected in this study to examine the effect of ground motion characteristics on its seismic response: the 1940 Imperial Valley, the 1995 Kobe, and the 1971 San Fernando earthquake. The baseline tank model is modified with different levels of water elevation (FR = 50%, 70%, and 90%), tank height (75%, 100%, and 125% of the baseline tank height), and tank diameter (80%, 100%, and 120% of the baseline tank diameter) to examine the

7. Conclusions In this research, the nonlinear characteristics of hydrodynamic pressure responses acting on the inner wall of cylindrical liquid storage tanks are examined using computational fluid dynamics (CFD). For this

(a) FR = 50%

Tank type

(b) FR = 70%

(c) FR = 90%

Fig. 17. Maximum hydrodynamic pressure distribution with different levels of water elevation under the San Fernando earthquake. 12

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(b) H=1.00H base

(a) H=0.75H base

(c) H=1.25H base

Fig. 18. PSD of hydrodynamic pressure response on the north side of the wall with different tank heights under the Imperial Valley earthquake.

(b) H=1.00H base

(a) H=0.75H base

(c) H=1.25H base

Fig. 19. PSD of hydrodynamic pressure response on the west side of the wall with different tank heights under the Kobe earthquake.

(b) H=1.00H b a s e

(a) H=0.75H base

(c) H=1.25H base

Fig. 20. PSD of hydrodynamic pressure response on the west side of the wall with different tank heights under the San Fernando earthquake. Table 4 Peak hydrodynamic wall pressure (kPa) under L + T + V components with different levels of tank diameter. Earthquake

Tank type

Normalized tank height (α ) 0.0

0.2

0.4

0.6

Imperial Valley

D = 0.8Dbase D = 1.0Dbase D = 1.2Dbase

241.0 309.7 297.8

231.2 299.2 292.1

213.1 268.0 276.3

186.1 221.5 239.1

Kobe

D = 0.8Dbase D = 1.0Dbase D = 1.2Dbase

169.4 228.0 259.6

144.3 209.5 210.9

132.1 187.7 174.3

116.1 156.9 137.8

San Fernando

D = 0.8Dbase D = 1.0Dbase D = 1.2Dbase

266.9 333.0 423.5

243.3 285.2 365.0

271.1 313.0 381.3

285.2 338.9 388.4

•

effect of these parameters on the dynamic fluid response in tanks under three orthogonal components of three earthquakes. The following conclusions from the parametric study can be drawn:

• Presence of vertical ground motion component: The pressure

response on the tank wall is significantly dependent on the ground 13

motion characteristics and the presence of vertical ground motion component. The hydrodynamic pressure energy near the surface of water under the combined action of two horizontal and one vertical ground motion components is much higher than that under two horizontal ground motion components. This is due to a significant change in flow for the vertical movement of water (vertical sloshing) induced by the vertical ground motion component. The particles of water from which are separated fall back to the water level result in the increase of the energy of hydrodynamic pressure near the water surface. In addition, the inclusion of vertical ground motion can change the response characteristics; the maximum pressure and the maximum total pressure distribution along the tank height as a result of the increase of the impulsive component energy of the fluid; and the location on the circumference where the peak total pressure occurs because of the Coriolis effects resulting from the combined ground motion components. Effect of water elevation level: For all three earthquakes, as the water elevation increases, the hydrodynamic pressure near the surface of water increases due to the increase of fluid mass. This is associated with the fact that as the water level decreases, an amount of water into which the air is entrained increases due to the excitations, and thus the fluid mass affecting the pressure on the wall decreases. In contrast, as the water level increases, the fluid mass

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(a) D=0.8D base

(b) D=1.0D base

(c) D=1.2D base

Fig. 21. PSD of hydrodynamic pressure response on the north side of the wall with different tank diameters under the Imperial Valley earthquake.

(a) D=0.8D base

(b) D=1.0D base

(c) D=1.2D base

Fig. 22. PSD of hydrodynamic pressure response on the west side of the wall with different tank diameters under the Kobe earthquake.

(a) D=0.8D base

(b) D=1.0D base

(c) D=1.2D base

Fig. 23. PSD of hydrodynamic pressure response on the west side of the wall with different tank diameters under the San Fernando earthquake.

•

increase of internal hydrodynamic pressure. Especially, the Imperial Valley earthquake and the San Fernando earthquake more significantly increase the pressure energy at the high frequency range (impulsive component), while the Kobe earthquake more considerably increases the pressure energy at the low frequency range (convective component).

contributing to the pressure acting on the wall increases. However, the water elevation differently affects the hydrodynamic pressure response according to the type of input earthquake excitation. For the Imperial earthquake, the extent of water elevation changes the location on the circumference at which the peak pressure due to the Coriolis effects. The Kobe earthquake does not change the position on the circumference where the peak hydrodynamic pressure occurs, yields a similar trend of the maximum hydrodynamic pressure distribution over the tank height regardless of the extent of water elevation, and increases the convective component rather than the impulsive component. Under the San Fernando earthquake, the peak hydrodynamic pressure typically occurs on the east side of the wall irrespective of the extent of water elevation, but the trend of the maximum hydrodynamic pressure distribution over the tank height varies from water elevation to water elevation. Overall the tank with FR = 90% exhibits the highest peak hydrodynamic pressure among three cases of water elevation for all three earthquakes. Effect of tank dimensions: As the tank height or diameter increases, the peak hydrodynamic pressure increases and the energy of dynamic pressure increases over the entire range of frequencies. It can be explained that the increase of liquid mass results in the

The results of the parametric study presented in this study underscore the need to appropriately account for the presence of vertical ground motion component, water elevation, and tank dimensions (height and diameter) in addition to the earthquake type. However, this study assumes the wall of tanks as rigid to compute the hydrodynamic pressure on their wall due to the nonlinear flow of fluid. If the wall is assumed to be flexible, the direction of the excitation is the same as that of the wall deformation and the fluid pressure acting on the flexible wall might be smaller. On the other hand, if the direction of the excitation is opposite to that of the wall movement, the fluid wall pressure becomes larger. Also, the wall pressure responses for the flexible wall can vary depending on several earthquake-related parameters such as seismic intensity, duration, and direction. Thus, further research is needed to study the flexible nonlinear fluid sloshing phenomenon 14

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(flexible wall) and to account for fluid-structure interaction in order to more realistically capture the seismic response of fluid and structure.

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